Algebraic Methods in Nonlinear Perturbation Theory

1. Front Matter

   Pages i-xii

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2. Matrix Perturbation Theory

   • V. N. Bogaevski, A. Povzner

   Pages 1-15

3. Systems of Ordinary Differential Equations with a Small Parameter

   • V. N. Bogaevski, A. Povzner

   Pages 17-45

4. Examples

   • V. N. Bogaevski, A. Povzner

   Pages 47-126

5. Reconstruction

   • V. N. Bogaevski, A. Povzner

   Pages 127-193

6. Equations in Partial Derivatives

   • V. N. Bogaevski, A. Povzner

   Pages 195-258

7. Back Matter

   Pages 259-266

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Many books have already been written about the perturbation theory of differential equations with a small parameter. Therefore, we would like to give some reasons why the reader should bother with still another book on this topic. Speaking for the present only about ordinary differential equations and their applications, we notice that methods of solutions are so numerous and diverse that this part of applied mathematics appears as an aggregate of poorly connected methods. The majority of these methods require some previous guessing of a structure of the desired asymptotics. The Poincare method of normal forms and the Bogolyubov-Krylov­ Mitropolsky averaging methods, well known in the literature, should be mentioned specifically in connection with what will follow. These methods do not assume an immediate search for solutions in some special form, but make use of changes of variables close to the identity transformation which bring the initial system to a certain normal form. Applicability of these methods is restricted by special forms of the initial systems.

• algebra
• applied mathematics
• bifurcation
• differential equation
• eigenvalue
• electromagnetic field
• field
• manifold
• matrix
• ordinary differential equation
• transformation

• Institute of the Physics of the Earth, Academy of Science—USSR, Moscow, USSR

  A. Povzner

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